3.135 \(\int \frac{\log (d (e+f \sqrt{x})^k) (a+b \log (c x^n))}{x^{3/2}} \, dx\)

Optimal. Leaf size=199 \[ \frac{4 b f k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{e}-\frac{2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{\sqrt{x}}+\frac{f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{2 f k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{4 b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{\sqrt{x}}-\frac{b f k n \log ^2(x)}{2 e}+\frac{2 b f k n \log (x)}{e}-\frac{4 b f k n \log \left (e+f \sqrt{x}\right )}{e}+\frac{4 b f k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e} \]

[Out]

(-4*b*f*k*n*Log[e + f*Sqrt[x]])/e - (4*b*n*Log[d*(e + f*Sqrt[x])^k])/Sqrt[x] + (4*b*f*k*n*Log[e + f*Sqrt[x]]*L
og[-((f*Sqrt[x])/e)])/e + (2*b*f*k*n*Log[x])/e - (b*f*k*n*Log[x]^2)/(2*e) - (2*f*k*Log[e + f*Sqrt[x]]*(a + b*L
og[c*x^n]))/e - (2*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/Sqrt[x] + (f*k*Log[x]*(a + b*Log[c*x^n]))/e +
(4*b*f*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/e

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Rubi [A]  time = 0.170308, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2454, 2395, 36, 29, 31, 2376, 2394, 2315, 2301} \[ \frac{4 b f k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{e}-\frac{2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{\sqrt{x}}+\frac{f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{2 f k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{4 b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{\sqrt{x}}-\frac{b f k n \log ^2(x)}{2 e}+\frac{2 b f k n \log (x)}{e}-\frac{4 b f k n \log \left (e+f \sqrt{x}\right )}{e}+\frac{4 b f k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e} \]

Antiderivative was successfully verified.

[In]

Int[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^(3/2),x]

[Out]

(-4*b*f*k*n*Log[e + f*Sqrt[x]])/e - (4*b*n*Log[d*(e + f*Sqrt[x])^k])/Sqrt[x] + (4*b*f*k*n*Log[e + f*Sqrt[x]]*L
og[-((f*Sqrt[x])/e)])/e + (2*b*f*k*n*Log[x])/e - (b*f*k*n*Log[x]^2)/(2*e) - (2*f*k*Log[e + f*Sqrt[x]]*(a + b*L
og[c*x^n]))/e - (2*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/Sqrt[x] + (f*k*Log[x]*(a + b*Log[c*x^n]))/e +
(4*b*f*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/e

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}} \, dx &=-\frac{2 f k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+\frac{f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-(b n) \int \left (-\frac{2 f k \log \left (e+f \sqrt{x}\right )}{e x}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x^{3/2}}+\frac{f k \log (x)}{e x}\right ) \, dx\\ &=-\frac{2 f k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+\frac{f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+(2 b n) \int \frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x^{3/2}} \, dx-\frac{(b f k n) \int \frac{\log (x)}{x} \, dx}{e}+\frac{(2 b f k n) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{e}\\ &=-\frac{b f k n \log ^2(x)}{2 e}-\frac{2 f k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+\frac{f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+(4 b n) \operatorname{Subst}\left (\int \frac{\log \left (d (e+f x)^k\right )}{x^2} \, dx,x,\sqrt{x}\right )+\frac{(4 b f k n) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{e}\\ &=-\frac{4 b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{\sqrt{x}}+\frac{4 b f k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e}-\frac{b f k n \log ^2(x)}{2 e}-\frac{2 f k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+\frac{f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+(4 b f k n) \operatorname{Subst}\left (\int \frac{1}{x (e+f x)} \, dx,x,\sqrt{x}\right )-\frac{\left (4 b f^2 k n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{e}\\ &=-\frac{4 b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{\sqrt{x}}+\frac{4 b f k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e}-\frac{b f k n \log ^2(x)}{2 e}-\frac{2 f k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+\frac{f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{4 b f k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{e}+\frac{(4 b f k n) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sqrt{x}\right )}{e}-\frac{\left (4 b f^2 k n\right ) \operatorname{Subst}\left (\int \frac{1}{e+f x} \, dx,x,\sqrt{x}\right )}{e}\\ &=-\frac{4 b f k n \log \left (e+f \sqrt{x}\right )}{e}-\frac{4 b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{\sqrt{x}}+\frac{4 b f k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e}+\frac{2 b f k n \log (x)}{e}-\frac{b f k n \log ^2(x)}{2 e}-\frac{2 f k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+\frac{f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{4 b f k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{e}\\ \end{align*}

Mathematica [A]  time = 0.396121, size = 145, normalized size = 0.73 \[ -\frac{4 b f k n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )}{e}-\frac{2 \left (a+b \log \left (c x^n\right )+2 b n\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{\sqrt{x}}-\frac{2 f k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)+2 b n\right )}{e}-\frac{f k \log (x) \left (-2 \left (a+b \log \left (c x^n\right )+2 b n\right )+4 b n \log \left (\frac{f \sqrt{x}}{e}+1\right )+b n \log (x)\right )}{2 e} \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^(3/2),x]

[Out]

(-2*Log[d*(e + f*Sqrt[x])^k]*(a + 2*b*n + b*Log[c*x^n]))/Sqrt[x] - (2*f*k*Log[e + f*Sqrt[x]]*(a + 2*b*n - b*n*
Log[x] + b*Log[c*x^n]))/e - (f*k*Log[x]*(4*b*n*Log[1 + (f*Sqrt[x])/e] + b*n*Log[x] - 2*(a + 2*b*n + b*Log[c*x^
n])))/(2*e) - (4*b*f*k*n*PolyLog[2, -((f*Sqrt[x])/e)])/e

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c{x}^{n} \right ) )\ln \left ( d \left ( e+f\sqrt{x} \right ) ^{k} \right ){x}^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)/x^(3/2),x)

[Out]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)/x^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, b f k n \log \left (x\right ) + b f k \log \left (c\right ) \log \left (x\right ) + a f k \log \left (x\right ) + \frac{b f k \log \left (x^{n}\right )^{2}}{2 \, n}}{e} - \frac{\frac{2 \,{\left (3 \, b f^{4} k x^{2} \log \left (x^{n}\right ) +{\left (3 \, a f^{4} k +{\left (4 \, f^{4} k n + 3 \, f^{4} k \log \left (c\right )\right )} b\right )} x^{2}\right )}}{\sqrt{x}} + \frac{18 \,{\left (b e^{4} x \log \left (x^{n}\right ) +{\left (a e^{4} +{\left (2 \, e^{4} n + e^{4} \log \left (c\right )\right )} b\right )} x\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k}\right )}{x^{\frac{3}{2}}} - \frac{9 \,{\left (b e f^{3} k x^{2} \log \left (x^{n}\right ) +{\left (a e f^{3} k +{\left (e f^{3} k n + e f^{3} k \log \left (c\right )\right )} b\right )} x^{2}\right )}}{x} + \frac{18 \,{\left ({\left (b e^{2} f^{2} k \log \left (c\right ) + a e^{2} f^{2} k\right )} x^{2} +{\left (a e^{4} \log \left (d\right ) +{\left (2 \, e^{4} n \log \left (d\right ) + e^{4} \log \left (c\right ) \log \left (d\right )\right )} b\right )} x +{\left (b e^{2} f^{2} k x^{2} + b e^{4} x \log \left (d\right )\right )} \log \left (x^{n}\right )\right )}}{x^{\frac{3}{2}}}}{9 \, e^{4}} + \int \frac{b f^{5} k x \log \left (x^{n}\right ) +{\left (a f^{5} k +{\left (2 \, f^{5} k n + f^{5} k \log \left (c\right )\right )} b\right )} x}{e^{4} f \sqrt{x} + e^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k)/x^(3/2),x, algorithm="maxima")

[Out]

integrate((b*f*k*x*log(x^n) + (a*f*k + (2*f*k*n + f*k*log(c))*b)*x)/x^2, x)/e - 1/9*(2*(3*b*f^4*k*x^2*log(x^n)
+ (3*a*f^4*k + (4*f^4*k*n + 3*f^4*k*log(c))*b)*x^2)/sqrt(x) + 18*(b*e^4*x*log(x^n) + (a*e^4 + (2*e^4*n + e^4*
log(c))*b)*x)*log((f*sqrt(x) + e)^k)/x^(3/2) - 9*(b*e*f^3*k*x^2*log(x^n) + (a*e*f^3*k + (e*f^3*k*n + e*f^3*k*l
og(c))*b)*x^2)/x + 18*((b*e^2*f^2*k*log(c) + a*e^2*f^2*k)*x^2 + (a*e^4*log(d) + (2*e^4*n*log(d) + e^4*log(c)*l
og(d))*b)*x + (b*e^2*f^2*k*x^2 + b*e^4*x*log(d))*log(x^n))/x^(3/2))/e^4 + integrate((b*f^5*k*x*log(x^n) + (a*f
^5*k + (2*f^5*k*n + f^5*k*log(c))*b)*x)/(e^4*f*sqrt(x) + e^5), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \sqrt{x} \log \left (c x^{n}\right ) + a \sqrt{x}\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k)/x^(3/2),x, algorithm="fricas")

[Out]

integral((b*sqrt(x)*log(c*x^n) + a*sqrt(x))*log((f*sqrt(x) + e)^k*d)/x^2, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))*ln(d*(e+f*x**(1/2))**k)/x**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right )}{x^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k)/x^(3/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d)/x^(3/2), x)